It is known how to find hyperbolic equations for asymptotics

The asymptotic equation is known: y= (b a)x (when the focus is on the x-axis), y= (a b)x

The focus is on the y-axis). The standard equation for the hyperbola: x a -y b now proves that the point on the hyperbola x a -y b = 1 is in the asymptote.

Let m(x,y) be the point at which the hyperbola is in the first quadrant, then.

y=（b/a）√（x²-a²）（x>a）

Because the curvilinear equation of x -a b>0) cofocal can be expressed as.

1( 0 is an ellipse, b2< So there are two asymptotes whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point where each branch reflects to form the other. In the case of the curve f(x)=1 x, the asymptote is two axes.

Hyperbolic equation x a -y b = 1

The asymptote equation simply replaces the 1 on the right with 0:

x²/a²-y²/b²=0

So: y b = x a

So: y= (b a)x

This is the asymptote equation.

Knowing the asymptote equation knows the value of b a;

Then you know the vertices of the hyperbolic equation and bring it in.

The asymptotic equation of the hyperbola: y= (b a)x (when the focus is on the x-axis), y= (a b)x (the focus is on the y-axis), or let the 1 in the hyperbolic standard equation x a -y b = 1 be zero, that is, the asymptotic equation is obtained.

The hyperbolic asymptotic equation is a geometric algorithm, which mainly solves the processing of some data of the building in practice. The main feature of the asymptote is that it is infinitely close, but cannot intersect.

It is divided into lead straight asymptotic, horizontal asymptote and oblique asymptote. It is an algorithm that is researched according to the actual needs of life.

1. The equation of the hyperbolic system co-asymptotic with the hyperbola x2 a2-y2 b2=1 (a 0, b 0) can be expressed as x2 a2-y2 b2= (0 and is a constant to be determined).

2. The curve system equation cofocal with the ellipse x2 a2-y2 b2=1 (a b 0) can be expressed as x2 a2-y2 b2=1 (the original ellipse is the original ellipse when =0 is the hyperbola when b2 a2).

The trajectory of the distance in the plane to the fixed point f(c,0) and the point to the point where the distance to the fixed line l:x=+(a2 c) is equal to the constant e=c a(c a 0), the fixed point is the focal point of the hyperbola, the fixed line is the alignment of the hyperbola, and the focal distance (focal parameter) p=a2 c, which is the same as the ellipse.

When the focus is on the x-axis, the asymptote of the hyperbola is y= (b a)*x, the hyperbolic equation.

is x 2 a 2-y 2 b 2 = 1, when the focus is on the y axis, the asymptote of the hyperbola is y = (a b)*x, and the hyperbolic equation is y 2 a 2-x 2 b 2 = 1

Asymptotics are divided into vertical asymptotic, horizontal, and oblique asymptotic.

It is important to note that not all curves have asymptotic lines, which reflect how some curves change as they extend indefinitely.

The asymptote of the hyperbola depends on the ratio of a and b, and when the focus is on the x-axis, the equation for the hyperbola asymptote is y= (b a)x when the focus is on the y-axis, the equation for the hyperbola asymptote is y=(a b)x

Therefore, the equation given for the hyperbola can uniquely determine the asymptote. Therefore, it is known that hyperbola is a sufficient condition for finding the asymptote.

However, the equation that only gives the asymptote cannot find the equation of the hyperbola. Since it is impossible to determine whether the focus is on the x-axis or y-axis according to the asymptote equation, it is impossible to know whether the slope of the asymptote is (b a) or (a b), so the equation that only gives the asymptote cannot find the hyperbolic equation. Therefore, it is known that hyperbola is not necessary to obtain the asymptote.

In summary, it is known that hyperbola is a sufficient but not necessary condition for obtaining the asymptote.

If you find it helpful, oh

Known equations for asymptotic equations: y= (b a)x (when the focus is on the x-axis), y= (a b)x (the focus is on the y-axis). The standard equation for hyperbola is obtained: x a -y b =1.

It is now proved that the point on the hyperbola x a -y b = 1 is in the asymptote.

Let m(x,y) be the point at which the hyperbola is in the first quadrant, then.

y=(b a) (x -a) filial piety and (x>a) Because x -a is y, the hyperbola points in the first quadrant are below the line y=bx a.

The focus is on the y-axisHyperbolaThe asymptotic equation is: y= (b a)x (when the focus is on the x-axis), y= (a b)x (the focus is on the y-axis) or let the 1 in the hyperbolic standard equation x 2 a 2-y 2 b 2 =1 be zero to get the asymptotic equation.

Simple geometric properties of hyperbola x 2 a 2-y 2 b 2 = 1:

1. Scope: |x|≥a,y∈r。

2. Symmetry: The symmetry of the hyperbola is exactly the same as that of the ellipse, and the center symmetry of the x-axis, y-axis and the origin of the stupid pin.

3. Vertices: two vertices a1 (-a, 0), a2 (a, 0), the line segment between the two vertices is the real axis, the length is 2a, the length of the imaginary axis is 2b, and c 2 = a 2 + b 2, which is different from the ellipse.

Asymptote features:

Infinitely close, but not intersecting. It is divided into vertical asymptotic, horizontal asymptote and oblique asymptotic.

When a point m on a curve is infinitely farther away from the origin along the curve, if the distance from m to a straight line is infinitely close to zero, then the straight line is called the asymptote of the curve.

It's important to note that not all curves have asymptottoes, which reflect how some curves change as they extend indefinitely. According to the position of the asymptote, the root plexus can be divided into three categories: horizontal asymptote, vertical asymptote, and oblique asymptote.

Just set it directly according to the first formula, so that the "1" on the right side of the first formula is equal to zero, which is the asymptote, that is, the second formula. I'll be next. This is the case where the focus is on the x-axis. If the focus is on the y-axis, the principle is the same.

bai my du adoption zhi rate].

If you don't understand, please keep asking! dao

If you still have new questions, you can continue to ask me for help a second time after you adopt it!

Your adoption is my driving force

o( o, remember to praise and adopt, help each other.

Good luck with your studies!

If you don't understand it yet, you can check the knowledge of conjugate hyperbola, and you should understand.